# High Dynamic Weak Signal Tracking Algorithm of a Beidou Vector Receiver Based on an Adaptive Square Root Cubature Kalman Filter

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## Abstract

**:**

## 1. Introduction

## 2. Vector Tracking Loop Operation Mechanism

## 3. Realization of Adaptive Square Root Cubature Kalman Filter Algorithm in Vector Receiver

#### 3.1. Cubature Kalman Filter

- (1)
- Prediction
- Initializing state quantity, error covariance, process noise, and measurement noise;
- Calculate and propagate volume points;
- Calculate the predicted value of states and error covariance.

- (2)
- Amendment
- Calculate and propagate volume point;
- Calculate the predicted value of the measurement;
- Calculate measurement error covariance and cross-covariance;
- Gain update, states, and error covariance update.

#### 3.2. Square Root Cubature Kalman Filter

- Time updating

- (1)
- Calculating and propagating cubature points

- (2)
- Calculating the predicted value of states

- (3)
- Calculating the square root of the covariance matrix of the prediction error ${\mathit{W}}_{k}$

- 2
- Measurement update

- (1)
- Calculating and propagating volume points

- (2)
- Calculating the measured predicted value

- (3)
- Calculating the square root of the innovation covariance matrix

- (4)
- Calculating the square root of the cross-covariance matrix

- (5)
- Estimating filter gain

- (6)
- Status update

- (7)
- Calculating the square root factor of the state estimation error covariance matrix

#### 3.3. Adaptive Square Root Cubature Kalman Filter Algorithm

#### 3.4. Realization of ASRCKF Algorithm in Vector Tracking Loop

#### 3.4.1. ASRCKF Vector Tracking Algorithm Implementation Specific Process

#### 3.4.2. The Specific Implementation Steps of ASRCKF

- Time update

- (1)
- First, the SRCKF algorithm is used to obtain the state estimation value ${\widehat{x}}_{k-1|k-1}$ and the square root of the state estimation error covariance matrix ${S}_{k-1|k-1}$ through the given model.

- (2)
- Calculate the predicted value of the state ${\widehat{x}}_{k|k-1}$ and ${S}_{k|k-1}$, which is the square root of the covariance matrix of the prediction error ${\mathit{W}}_{k}$.

- 2.
- Measurement update
- (1)
- The measured predicted value is obtained through Equations (24)–(26).
- (2)
- ${S}_{zz,k|k-1}$, the real-time square root of the innovation covariance matrix is obtained through Equation (30).
- (3)
- The real-time updated value of the state and square root factor of the state estimation error covariance matrix is obtained through Equations (22)–(26), and then enters the next cycle.

## 4. Simulation Test

_{0}= 21.5, the tracking accuracy of the ASRCKF VLL is 6 dB higher than that of the SRCKF VLL and 10 dB higher than that of the ASRCKF SLL. When C/N

_{0}= 22.5, the mean square error of the frequency estimation of the ASRCKF VLL is 31 Hz, which is 5 dB higher than that of the SRCKF VLL and 9 dB higher than that of the ASRCKF SLL, but as the carrier-to-noise ratio C/N

_{0}increases, this improvement gradually decreases.

## 5. Algorithm Performance Analysis

_{0}= 30 dB-Hz.

_{0}is 25, 23, 22, 21, and 19 dB-Hz, respectively, can be obtained. Table 2 describes the comparison of the filtering performance of the SRCKF and ASRCKF algorithms when the C/N

_{0}is 25, 23, 21, 20, and 19 dB-Hz, respectively.

## 6. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Glossary

List of symbols | |

$\mathit{V}$ | velocity vector |

${\mathit{V}}^{\prime}$ | acceleration vector |

${\mathit{V}}^{\u2033}$ | Jerk vector |

${\mathit{X}}_{k}$ | state, ‘k’ represents the time epoch of the data sample |

${\mathit{W}}_{k}$ | state noise |

${\mathit{Z}}_{k}$ | measurement |

${\mathit{U}}_{k}$ | measurement noise |

${z}_{code}$ | the output of code phase discriminator |

${z}_{carrier}$ | the output of carrier frequency discriminator |

${e}_{j,k}$ | line-of-sight direction unit vector between the receiver and the jth satellite at time epoch k |

${R}_{k}$ | measurement error covariance matrix |

${Q}_{k-1}$ | represents the covariance matrix of the prediction error ${\mathit{W}}_{k-1}$ |

${X}_{k|k-1}^{*}$ | represents the weighted center matrix, |

${S}_{zz}$ | square root of the innovation covariance matrix |

${S}_{Q,k}$ | square root factor of ${Q}_{k}$ |

${S}_{R,k}$ | square root factor of the measurement error covariance matrix ${R}_{k}$ |

${\gamma}_{k|k-1}$ | weighted center matrix |

${K}_{k}$ | filter gain |

${\widehat{z}}_{k|k-1}$ | measured predicted value |

${S}_{k|k}$ | square root factor of the state estimation error covariance matrix |

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**Figure 2.**The trajectory of the receiver relative to the Beidou satellite. (

**a**) Acceleration model for highly dynamic motion; (

**b**) Jerk model for highly dynamic motion. Reprinted from ref. [24].

**Figure 6.**$RMS{E}_{\mathrm{time}}$ value when the filter uses the ASRCKF algorithm and the SRCKF algorithm to filter respectively and C/N

_{0}= 30 dB-Hz.

Position Error—Variance (m ^{2}) | Velocity Error—Variance (m ^{2}/s^{2}) | Code Phase Error—Mean Square Error | Carrier Frequency Error—Mean Square Error |
---|---|---|---|

10 | 2 | 12° | 0.02 Chip |

**Table 2.**The average time consumption and filtering performance of the two algorithms of ASRCKF and SRCKF when the C/N

_{0}is 25, 23, 21, 20, and 19 dB-Hz, respectively.

Carrier-to-Noise Ratio (C/N0, dB/Hz) | Average Consumption Time of ASRCKF (s) | Average Consumption Time of SRCKF (s) | $\mathbf{Average}\mathbf{Value}\mathbf{of}\mathit{R}\mathit{M}\mathit{S}{\mathit{E}}_{\mathit{t}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}$ of ASRCKF (s) | $\mathbf{Average}\mathbf{Value}\mathbf{of}\mathit{R}\mathit{M}\mathit{S}{\mathit{E}}_{\mathit{t}\mathit{o}\mathit{t}\mathit{a}\mathit{l}}$ of SRCKF (s) |
---|---|---|---|---|

25 | 0.0122 | 0.0123 | 0.0125 | 0.025 |

23 | 0.015 | 0.015 | 0.019 | 0.026 |

21 | 0.019 | 0.021 | 0.022 | 0.03 |

20 | 0.022 | 0.025 | 0.025 | 0.039 |

19 | 0.025 | 0.025 | 0.029 | 0.05 |

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**MDPI and ACS Style**

Li, N.; Zhang, S.; Jiang, Y.
High Dynamic Weak Signal Tracking Algorithm of a Beidou Vector Receiver Based on an Adaptive Square Root Cubature Kalman Filter. *Sensors* **2021**, *21*, 6707.
https://doi.org/10.3390/s21206707

**AMA Style**

Li N, Zhang S, Jiang Y.
High Dynamic Weak Signal Tracking Algorithm of a Beidou Vector Receiver Based on an Adaptive Square Root Cubature Kalman Filter. *Sensors*. 2021; 21(20):6707.
https://doi.org/10.3390/s21206707

**Chicago/Turabian Style**

Li, Na, Shufang Zhang, and Yi Jiang.
2021. "High Dynamic Weak Signal Tracking Algorithm of a Beidou Vector Receiver Based on an Adaptive Square Root Cubature Kalman Filter" *Sensors* 21, no. 20: 6707.
https://doi.org/10.3390/s21206707